F.T. Farrell and P. Ontaneda plan to continue their study of the topology of the space of negatively curved metrics supported by a given smooth manifold; together with its relatives: the Teichmuller space of negatively curved metrics and the moduli space of negatively curved metrics. This is a new area of study for manifolds whose dimension is greater than two, and there are potentially many directions in which to go, and a large number of questions that can be addressed. Farrell and Ontaneda also plan to continue their research on the general area of manifold topology and its applications to geometry.
A Riemannian metric on a space is a rule that stipulates how to calculate angles and lengths of paths, and Riemannian Geometry is the study of these objects. An important set of metrics are those that are negatively curved, that is, metrics for which the sum of the angles of every triangle is always less than 180 degrees. The shape of a space (i.e. its Topology) places restrictions on the type of metrics it can support. For instance there are spaces that don't support a negatively curved metric. Examples of this type of space are the sphere and the surface of a (one-holed) doughnut. On the other hand the two-holed doughnut does support a negatively curved metric. For spaces of this sort the collection of all possible negatively curved metrics on it form the points of a new space whose shape (topology) is the object of study for this project.
on the NSF grant received during the years 2009-2012 with award number 51788. A Riemannian metric on a space is a rule that stipulates how to calculateangles and lengths of paths, and Riemannian Geometry is the study of theseobjects. An important set of metrics are those that are negatively curved,that is, metrics for which the sum of the angles of every triangle isalways less than 180 degrees. The shape of a space (i.e. its Topology) places restrictions on the typeof metrics it can support. For instance there are spaces that don'tsupport a negatively curved metric. Examples of this type of space are thesphere and the surface of a (one-holed) doughnut. On the other hand thetwo-holed doughnut does support a negatively curved metric. For spaces ofthis sort the collection of all possible negatively curved metrics on itform the points of a new space whose shape (topology) is the object ofstudy for this project. This is a new area of study for spaces whose dimension isgreater than two, and there are many directions in which togo, and a large number of questions that can be addressed. Farrell andOntaneda have answered many of these questions as a result of this project.
